$1.$List the following asymptotic complexities from best $($top$)$ to worst. Which functions are the same rank?

$($ n lg n $)$

$($ n$2)$

$($ lg n $)$

$($ nn $)$

$($ log n $)$

$($n $)$

$($ n $$n $)$

$($ lg lg n $)$

$($ n$!)$

$(1)$

$($ n lg n$2)$

$($ n lg$2$ n $)$

$($ n $)$

$($ n$2$ lg n $)$

$($ nlg n $)$

$($ n$3$ lg n $)$

$($ n$3)$

$2.$Rank these $1-5($Fastest $=1,$ Slowest $=5)$:

$\_\_\_\_($ nlg n $)$

$\_\_\_\_($ n$2$ lg n $)$

$\_\_\_\_($ n lg$2$ n $)$

$\_\_\_\_($ n lg n$2)$

$\_\_\_\_($ lgn n $)$

$\_\_\_\_($ lg nn $)$

$\_\_\_\_($ n lg n $)$

$3.$Rank these $1-7($Fastest $=1,$ Slowest $=7)$:

$\_\_\_\_($ nlg n $)$

$\_\_\_\_($ lgn n $)$

$\_\_\_\_($ lg nn $)$

$\_\_\_\_($ n lg n $)$

$\_\_\_\_($ n$2$ lg n $)$

$\_\_\_\_($ n lg$2$ n $)$

$\_\_\_\_($ n lg n$2)$

$4.$ Reduce these and list them smallest to largest. Note any equalities by listing them together as the same rank.

$($ n lg$2$ n $+$ n lg n$2)$

$($ nlg n $+$ n$2$ lg n $)$

$($ lgn n $+$ lg nn $)$

$($ n lg n $+$ nlg n$)$

$($ lg$2$ n $+3$n$)$